"However, while it may not be possible to declare war against "a group that is not a country", Iraq is a Country -- and the Constitution provides clear details on how the United States is to go about making War on a Country."
Then what about Afghanistan? It is a country too.
True, but Ron Paul voted for a Resolution advising the President to go after whomever was responsible for 9-11, since nobody was 100% certain exactly which state actors or non-state actors were directly responsible.
The President determined that the Taliban government of Afghanistan was one of the responsible parties, and so "went after" them as instructed by the Resolution. However, various Non-State actors were also involved; and as Ron Paul stated, it's difficult to imagine how to Declare War on a Non-Country. So, Ron Paul supported a Resolution which would authorize military force against Non-State actors as well.